Thoroughly whisk eggs, then add cream and vanilla. Heat 1 teaspoon of the oil and 1 teaspoon of the butter in a 10 to 12-inch cast iron skillet over medium heat.
Slowly dip a French bread slice in the egg mixture, turning to coat thoroughly. Place bread in preheated skillet and repeat with 2 more bread slices. Cook until golden brown and cooked through, turning once, about 3 minutes per side.
Carefully wipe out skillet, add another teaspoon each of the oil and butter, and repeat the process with three more bread slices.
Repeat the steps with the remaining oil, butter, and bread. Serve with maple syrup.